Problem 5 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\alpha, \beta\) are roots of the quadratic equation \(x^{2}\) \(+p x+p^{2}+q=0\), then prove that \(\alpha^{2}+\alpha \beta+\) \(\beta^{2}+q=0\)

Step-by-Step Solution

Answer

Using algebraic identities and Vieta's formulas, we show that \(\alpha^2 + \alpha\beta + \beta^2 + q = 0\).

Step 1: Understand the given quadratic equation

The quadratic equation given is \(x^2 + px + p^2 + q = 0\). In this equation, \(\alpha\) and \(\beta\) are the roots.

Step 2: Use Vieta's formulas

According to Vieta's formulas, for a quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\), we have:1. \(\alpha + \beta = -\frac{b}{a}\)2. \(\alpha\beta = \frac{c}{a}\)For the given equation:- \(\alpha + \beta = -p\)- \(\alpha\beta = p^2 + q\)

Step 3: Using algebraic identities

We need to show that \(\alpha^2 + \alpha\beta + \beta^2 + q = 0\). To do this, we first remember the identity:\[\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\]Substitute \(\alpha + \beta = -p\) and \(\alpha\beta = p^2 + q\) into this identity.

Step 4: Calculate \(\alpha^2 + \beta^2\)

Using the identity from step 3:\[\alpha^2 + \beta^2 = (-p)^2 - 2(p^2 + q)\]Simplifying this gives:\[\alpha^2 + \beta^2 = p^2 - 2p^2 - 2q = -p^2 - 2q\]

Step 5: Calculate \(\alpha^2 + \alpha\beta + \beta^2 + q\)

Using the expression for \(\alpha^2 + \beta^2\) from Step 4, we substitute:\[\alpha^2 + \alpha\beta + \beta^2 + q = (-p^2 - 2q) + (p^2 + q) + q\]Simplifying this expression results in:\[= -p^2 - 2q + p^2 + q + q = 0\]

Step 6: Conclusion: Prove the given expression

The simplification in Step 5 shows that \(\alpha^2 + \alpha\beta + \beta^2 + q = 0\) is satisfied, thus proving the required expression.

Key Concepts

Vieta's formulas

Vieta's formulas offer a simple way to relate the coefficients of a polynomial to sums and products of its roots. They are particularly useful for quadratic equations. For a quadratic equation of the form \(ax^2 + bx + c = 0\), these formulas state the following about the roots \(\alpha\) and \(\beta\):

• The sum of the roots, \(\alpha + \beta\), is given by \(-\frac{b}{a}\).
• The product of the roots, \(\alpha\beta\), is \(\frac{c}{a}\).

In the given exercise, for the equation \(x^2 + px + p^2 + q = 0\), the coefficients are \(a=1\), \(b=p\), and \(c=p^2 + q\). Applying Vieta's formulas helps find:

• \(\alpha + \beta = -p\)
• \(\alpha\beta = p^2 + q\)

These expressions are key in helping us manipulate and simplify the equations to ultimately prove the required identity. Understanding Vieta’s formulas is crucial when dealing with exercises that involve relationships between roots and coefficients.

Algebraic identities

Algebraic identities are powerful tools in simplifying and manipulating mathematical expressions to unveil hidden relationships. In this problem, the identity for the sum of squares comes into play:

• \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)

This identity is particularly useful here because it lets us rewrite \(\alpha^2 + \beta^2\) in terms of \(\alpha + \beta\) and \(\alpha\beta\), both of which we know from Vieta's formulas.By expressing \(\alpha^2 + \beta^2\) in this way and substituting \(\alpha + \beta = -p\) and \(\alpha\beta = p^2 + q\), we derive \[\alpha^2 + \beta^2 = (-p)^2 - 2(p^2 + q)\]. This identity fosters clarity and keeps equations manageable, allowing us to eventually demonstrate that \(\alpha^2 + \alpha\beta + \beta^2 + q = 0\). Mastering such identities can simplify complex transformations, inviting an understanding of deeper properties of equations.

Roots of equations

The concept of roots is foundational in algebra, especially for solving equations. Roots are the values for which a given equation holds true, i.e., they make the equation equal to zero. In the context of quadratic equations, such as \(x^2 + px + p^2 + q = 0\), finding the roots \(\alpha\) and \(\beta\) involves solving for \(x\) so that the equation resolution is zero.Vieta’s formulas aid in understanding the relationship between these roots and the coefficients. Once the roots are known or expressed in another form, such as using Vieta's formulas, we can use algebraic identities to explore further properties or prove new expressions.The task in the original problem is not about computing the roots directly but proving a deeper relationship involving them, \(\alpha^2 + \alpha\beta + \beta^2 + q = 0\), showcasing how even without explicitly finding \(\alpha\) and \(\beta\), their properties governed by the equation can be manipulated to derive new truths. Recognizing roots as solutions not only answers direct questions but also builds bridges to more profound insights into equations.